[1]} prove that if one is able to approximately solve a two-player zero-sum moment-matching game between a policy player (that picks from \(\Pi \) ) and a discriminator (that picks from \(\mathcal {F}_{Q_E}\) , \(\mathcal {F}_{Q}\) , or \(\mathcal {F}_{r}\) ), one has a bound on the performance difference between the learner and the expert. Depending on which class the discriminator selects from, one ends up with a different bound. For example, if one solves the game with the following payoff and \(f \in \mathcal {F}_{r}\)  to an \(\epsilon \) -approximate Nash equilibrium,
\(U(\pi , f) = \frac{1}{T}(\mathop {{}\mathbb {E}}_{\tau \sim \pi }[\sum _{t=1}^T f(s_t, a_t, c)] - \mathop {{}\mathbb {E}}_{\begin{array}{c}\tau \sim \pi ^E\end{array}}[\sum _{t=1}^T f(s_t, a, c)]),\) 
